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NAME
chgeqz - implement a single-shift version of the QZ method
for finding the generalized eigenvalues
w(i)=ALPHA(i)/BETA(i) of the equation det( A-w(i) B ) = 0
If JOB='S', then the pair (A,B) is simultaneously reduced to
Schur form (i.e., A and B are both upper triangular) by
applying one unitary tranformation (usually called Q) on the
left and another (usually called Z) on the right
SYNOPSIS
SUBROUTINE CHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, A, LDA,
B, LDB, ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK,
RWORK, INFO )
CHARACTER COMPQ, COMPZ, JOB
INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, LWORK, N
REAL RWORK( * )
COMPLEX A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), Q(
LDQ, * ), WORK( * ), Z( LDZ, * )
#include <sunperf.h>
void chgeqz(char job, char compq, char compz, int n, int
ilo, int ihi, complex *ca, int lda, complex *cb,
int ldb, complex *calpha, complex *cbeta, complex
*q, int ldq, complex *cz, int ldz, int *info);
PURPOSE
CHGEQZ implements a single-shift version of the QZ method
for finding the generalized eigenvalues
w(i)=ALPHA(i)/BETA(i) of the equation A are then
ALPHA(1),...,ALPHA(N), and of B are BETA(1),...,BETA(N).
If JOB='S' and COMPQ and COMPZ are 'V' or 'I', then the uni-
tary transformations used to reduce (A,B) are accumulated
into the arrays Q and Z s.t.:
Q(in) A(in) Z(in)* = Q(out) A(out) Z(out)*
Q(in) B(in) Z(in)* = Q(out) B(out) Z(out)*
Ref: C.B. Moler & G.W. Stewart, "An Algorithm for General-
ized Matrix
Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
pp. 241--256.
ARGUMENTS
JOB (input) CHARACTER*1
= 'E': compute only ALPHA and BETA. A and B will
not necessarily be put into generalized Schur
form. = 'S': put A and B into generalized Schur
form, as well as computing ALPHA and BETA.
COMPQ (input) CHARACTER*1
= 'N': do not modify Q.
= 'V': multiply the array Q on the right by the
conjugate transpose of the unitary tranformation
that is applied to the left side of A and B to
reduce them to Schur form. = 'I': like COMPQ='V',
except that Q will be initialized to the identity
first.
COMPZ (input) CHARACTER*1
= 'N': do not modify Z.
= 'V': multiply the array Z on the right by the
unitary tranformation that is applied to the right
side of A and B to reduce them to Schur form. =
'I': like COMPZ='V', except that Z will be ini-
tialized to the identity first.
N (input) INTEGER
The order of the matrices A, B, Q, and Z. N >= 0.
ILO (input) INTEGER
IHI (input) INTEGER It is assumed that A is
already upper triangular in rows and columns
1:ILO-1 and IHI+1:N. 1 <= ILO <= IHI <= N, if N >
0; ILO=1 and IHI=0, if N=0.
A (input/output) COMPLEX array, dimension (LDA, N)
On entry, the N-by-N upper Hessenberg matrix A.
Elements below the subdiagonal must be zero. If
JOB='S', then on exit A and B will have been
simultaneously reduced to upper triangular form.
If JOB='E', then on exit A will have been des-
troyed.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(
1, N ).
B (input/output) COMPLEX array, dimension (LDB, N)
On entry, the N-by-N upper triangular matrix B.
Elements below the diagonal must be zero. If
JOB='S', then on exit A and B will have been
simultaneously reduced to upper triangular form.
If JOB='E', then on exit B will have been des-
troyed.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(
1, N ).
ALPHA (output) COMPLEX array, dimension (N)
The diagonal elements of A when the pair (A,B) has
been reduced to Schur form. ALPHA(i)/BETA(i)
i=1,...,N are the generalized eigenvalues.
BETA (output) COMPLEX array, dimension (N)
The diagonal elements of B when the pair (A,B) has
been reduced to Schur form. ALPHA(i)/BETA(i)
i=1,...,N are the generalized eigenvalues. A and
B are normalized so that BETA(1),...,BETA(N) are
non-negative real numbers.
Q (input/output) COMPLEX array, dimension (LDQ, N)
If COMPQ='N', then Q will not be referenced. If
COMPQ='V' or 'I', then the conjugate transpose of
the unitary transformations which are applied to A
and B on the left will be applied to the array Q
on the right.
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= 1.
If COMPQ='V' or 'I', then LDQ >= N.
Z (input/output) COMPLEX array, dimension (LDZ, N)
If COMPZ='N', then Z will not be referenced. If
COMPZ='V' or 'I', then the unitary transformations
which are applied to A and B on the right will be
applied to the array Z on the right.
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >= 1.
If COMPZ='V' or 'I', then LDZ >= N.
WORK (workspace/output) COMPLEX array, dimension
(LWORK)
On exit, if INFO >= 0, WORK(1) returns the optimal
LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >=
max(1,N).
RWORK (workspace) REAL array, dimension (N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille-
gal value
= 1,...,N: the QZ iteration did not converge.
(A,B) is not in Schur form, but ALPHA(i) and
BETA(i), i=INFO+1,...,N should be correct. =
N+1,...,2*N: the shift calculation failed. (A,B)
is not in Schur form, but ALPHA(i) and BETA(i),
i=INFO-N+1,...,N should be correct. > 2*N:
various "impossible" errors.
FURTHER DETAILS
We assume that complex ABS works as long as its value is
less than overflow.
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